Generic measures for geodesic flows on nonpositively curved manifolds

Yves Coudène, Barbara Schapira

Abstract

We study the generic invariant probability measures for the geodesic flow
on connected complete nonpositively curved manifolds.
Under a mild technical assumption,
we prove that ergodicity is a generic property in the set of
probability measures defined on the unit tangent bundle of the manifold
and supported by trajectories not bounding a flat strip.
This is done by showing that Dirac measures on periodic orbits
are dense in that set.

In the case of a compact surface, we get the following sharp result:
ergodicity is a generic property in the space of all invariant measures
defined on the unit tangent bundle of the surface
if and only if there are no flat strips in the universal cover of the surface.

Finally, we show under suitable assumptions that generically, the invariant
probability measures have zero entropy and are not strongly mixing.
1

Ergodicity is a generic property in the space of probability measures
invariant by a topologically mixing Anosov flow on a compact manifold.
This result, proven by K. Sigmund in
the seventies [Si72],
implies that on a compact connected negatively curved manifold,
the set of ergodic measures is a dense
Gδ subset of the set of all probability measures invariant by the
geodesic flow. The proof of K. Sigmund’s result is based on the specification
property. This property relies on the uniform hyperbolicity of the system and
on the compactness of the ambient space.

In [CS10], we showed that ergodicity is a generic property of
hyperbolic systems without relying on the specification property.
As a result, we were able to prove that the set of
ergodic probability measures invariant by the geodesic flow, on a negatively
curved manifold, is a dense Gδ set, without any compactness
assumptions or pinching assumptions on the sectional curvature of the
manifold.

A corollary of our result is the existence of ergodic invariant probability
measures of full support for the geodesic flow on any complete negatively
curved manifold, as soon as the flow is transitive. Surprisingly, we succeeded
in extending this corollary to the nonpositively curved setting.
However, the question of genericity in nonpositive curvature appears to be
much more difficult, even for surfaces. In [CS11], we gave examples of
compact nonpositively curved surfaces with negative Euler characteristic
for which ergodicity is not a generic property in
the space of probability measures invariant by the geodesic flow.

The first goal of the article is to obtain genericity results in the non
positively curved setting. From now on, all manifolds are assumed to be
connected, complete Riemannian manifolds. Recall that a flat
strip in the universal cover of the manifold is a totally geodesic subspace
isometric to the space [0,r]×R, for some r>0, endowed with its
standard euclidean structure.
We first show that if there are no flat strip, genericity holds.

Theorem 1.1

Let M be a nonpositively curved manifold,
such that its universal cover has no flat strips.
Assume that the geodesic flow has at least
three periodic orbit on the unit tangent bundle T1M of M. Then
the set of ergodic probability measures on T1M is a dense Gδ-subset
of the set of all probability measures invariant by the flow.

This theorem is a particular case of theorem 1.3 below.
In the two-dimensional compact case, we get the following sharp result.

Theorem 1.2

Let M be a nonpositively curved compact orientable surface, with negative
Euler characteristic. Then ergodicity is a generic property in the set
of all invariant probability measures on T1M if and only if
there are no
flat strips on the universal cover of M.

In the higher dimensional case, the situation is more complicated.
Under some technical assumption,
we prove that genericity holds in restriction to the set of
nonwandering vectors whose lifts do not bound a flat strip.

Theorem 1.3

Let M be a connected, complete, nonpositively curved manifold,
and T1M its unit tangent bundle.
Denote by Ω⊂T1M the nonwandering set of the
geodesic flow, and ΩNF⊂Ω the set of nonwandering
vectors that do not bound a flat strip.
Assume that ΩNF is open in Ω, and contains at least three
different periodic orbits of the geodesic flow.

Then the set of ergodic probability measures invariant by the
geodesic flow and with full support in ΩNF is a Gδ-dense
subset of the set of invariant probability measures on ΩNF.

The assumption that ΩNF is open in Ω is satisfied
in many examples. For instance, it is true as soon as the number
of flat strips on the manifold is finite.
The set of periodic orbits of the geodesic flow is in 1−1-correspondence
with the set of oriented closed geodesics on the manifold. Thus, the
assumption that ΩNF contains at least three different periodic
orbits means that there are at least two distinct nonoriented
closed geodesics in the manifold that do not lie in the projection
of a flat strip.
This assumption rules out a few uninteresting examples, such as
simply connected manifolds or cylinders, and corresponds to the
classical assumption of nonelementaricity in negative curvature.

Whether ergodicity is a generic property in the space of all invariant
measures, in presence of flat strips of intermediate dimension, is still an
open question. In section 4.4, we will see examples with periodic
flat strips of maximal dimension where ergodicity is not generic.

The last part of the article is devoted to mixing and entropy.
Inspired by results of [ABC10], we
study the genericity of other dynamical properties of measures,
as zero entropy or mixing. In particular, we prove that

Theorem 1.4

Let M be a connected, complete, nonpositively curved manifold,
such that ΩNF contains
at least three different periodic orbits of the geodesic flow and
is open in the nonwandering set Ω.

The set of invariant probability measures with zero entropy
for the geodesic flow is generic in the set of invariant
probability measures on ΩNF.
Moreover, the set of invariant probability measures on ΩNF
that are not strongly mixing is a generic set.

The assumptions in all our results include the case where M
is a noncompact negatively curved manifold.
In this situation, we have Ω=ΩNF.
Even in this case, theorem 1.4 is new.
When M is a compact negatively curved manifold, it follows
from [Si72], [Pa62].
Theorem 1.3 was proved in [CS10] in the negative
curvature case.

Results above show that under our assumptions, ergodicity is generic, and
strong mixing is not. We don’t know under which condition weak-mixing is a
generic property, except for compact negatively curved manifolds [Si72].
In contrast, topological mixing holds most of the time, and is equivalent to
the non-arithmeticity of the length spectrum (see proposition
6.2).

In section 2, we recall basic facts on nonpositively curved
manifolds and define interesting invariant sets for the geodesic flow.
In section 3, we study the case of surfaces.
The next section is devoted to the proof
of theorem 1.3.
At last, we prove theorem 1.4 in
sections 5 and 6.

During this work, the authors benefited from the ANR grant ANR-JCJC-0108 Geode.

Let M be a Riemannian manifold with nonpositive curvature, and let v be a
vector belonging to the unit tangent bundle T1M of M. The vector v is a
rank one vector, if the only parallel Jacobi fields along the geodesic
generated by v are proportional to the generator of the geodesic flow. A
connected complete nonpositively curved manifold is a rank one
manifold if its tangent bundle admits a rank one vector. In that case, the
set of rank one vectors is an open subset of T1M. Rank one vectors
generating closed geodesics are precisely the hyperbolic periodic points of
the geodesic flow. We refer to the survey of G. Knieper [K02] and the
book of W. Ballmann [Ba95]
for an overview of the properties of rank one manifolds.

Let X⊂T1M be an invariant set under the action
of the geodesic flow (gt)t∈R.
Recall that the strong stable sets of the flow on X are defined by :

Wss(v):={w∈X|limt→∞d(gt(v),gt(w))=0} ;
Wssε(v):={w∈Wss(v)|d(gt(v),gt(w))≤ε for all t≥0}.

One also defines the strong unstable sets Wsu and
Wsuε of gt ; these are the stable sets of
g−t.

Denote by Ω⊂T1M the
nonwandering set of the geodesic flow, that
is the set of vectors v∈T1M such that
for all neighbourhoods V of v,
there is a sequence tn→∞ ,
with gtnV∩V≠∅.
Let us introduce several interesting invariant subsets
of the nonwandering set Ω of the geodesic flow.

Definition 2.1

Let v∈T1M.
We say that its strong stable (resp. unstable) manifold coincides
with its strong stable (resp. unstable) horosphere if, for any lift
~v∈T1~M of v,
for all ~w∈T1~M,
the existence of a constant C>0 s.t.
d(gt~v,gt~w)≤C for
all t≥0
(resp. t≤0) implies that there exists τ∈R such that
d(gtgτ~v,gt~w)→0
when t→+∞
(resp. t→−∞).

The terminology comes from the fact that on Ωhyp, a lot of
properties of a hyperbolic flow still hold. However, periodic orbits in
Ωhyp are not necessarily hyperbolic in the sense that they can have
zero Lyapounov exponents, for example higher rank periodic vectors.

Definition 2.2

Let v∈T1M.
We say that v does not bound a flat strip
if no lift ~v∈T1~M of v determines a geodesic
which bounds an infinite flat (euclidean) strip isometric
to [0,r]×R, r>0, on T1~M.

The projection of a flat strip on the manifold M
is called a periodic flat strip if it contains a periodic geodesic.

We say that v is not contained in a periodic flat strip
if the geodesic determined by v on M does not stay
in a periodic flat strip for all t∈R.

In [CS10], we restricted the study of the dynamics to the set Ω1 of
nonwandering rank one vectors whose stable (resp. unstable) manifold coincides
with the stable (resp. unstable) horosphere. If R1 denotes
the set of rank one vectors, then Ω1=Ωhyp∩R1.
The dynamics on Ω1 is very close from the dynamics
of the geodesic flow on a negatively curved manifold,
but this set is not very natural, and too small in general.
We improve below our previous results,
by considering the following larger sets:

the set ΩNF of nonwandering vectors that do not bound a flat strip,

the set ΩNFP of nonwandering vectors that are not contained in
a periodic flat strip,

the set Ωhyp of nonwandering vectors
whose stable (resp. unstable) manifold coincides with the stable horosphere.

We have the inclusions

Ω1⊂Ωhyp⊂ΩNF⊂ΩNFP⊂Ω,

and they can be strict, except if M has negative curvature, in which case
they all coincide. Indeed, a higher rank periodic vector is not in Ω1,
but it can be in Ωhyp when it does not bound a flat strip of
positive width. A rank one vector whose geodesic is asymptotic to a flat
cylinder is in ΩNF but not in Ωhyp.

Question 2.3

It would be interesting to understand when we have the equality
ΩNF=ΩNFP.
We will show that on
compact rank one surfaces, if there is a flat strip, then there exists also a periodic flat strip.
When the surface is a flat torus,
we have of course ΩNF=ΩNFP=∅.

It could also happen on some noncompact rank one manifolds
that all vectors that bound a nonperiodic flat
strip are wandering, so that ΩNF=ΩNFP.

Is it true on all rank-one surfaces, and/or all rank-one compact manifolds,
that ΩNF=ΩNFP ?

In the negative curvature case, it is standard to assume the fundamental group of M to be nonelementary. This means
that there exists at least two (and therefore infinitely many)
closed geodesics on M, and therefore at least four (and in fact
infinitely many) periodic orbits of the geodesic flow
on T1M (each closed geodesic lifts to T1M into
two periodic curves, one for each orientation).
This allows to discard simply connected manifolds or hyperbolic cylinders,
for which there is no interesting recurring dynamics.

In the nonpositively curved case, we must also get rid of
flat euclidean cylinders, for which there are infinitely many
periodic orbits, but no other recurrent trajectories.
So we will assume that there exist at least three different periodic
orbits in ΩNF, that is, two distinct closed
geodesics on M that do not bound a flat strip.

We will need another stronger assumption, on the flats of the manifold.
To avoid to deal with flat strips, we will
work in restriction to ΩNF,
with the additional assumption that ΩNF is open in Ω.
This is satisfied for example if M admits
only finitely many flat strips.
We will see that this assumption insures that the periodic orbits
that do not bound a flat strip are dense in Ωhyp and ΩNF.

In the proof of theorems 1.3
and 1.4, the key step is the proposition below.

Proposition 2.4

Let M be a connected, complete, nonpositively curved manifold,
which admits at least three different periodic orbits that do not
bound a flat strip. Assume
that ΩNF is open in Ω.
Then the Dirac measures supported by the periodic orbits
of the geodesic flow (gt)t∈R that are in ΩNF,
are dense in the set of all invariant probability measures defined
on ΩNF.

In this section, M is a compact, connected, nonpositively curved
orientable surface. We prove theorem 1.2.

If the surface admits a periodic flat strip, by our results in [CS11],
we know that ergodicity cannot be generic.
In particular, a periodic orbit in the middle of the flat strip is not
in the closure of any ergodic invariant probability measure of full support.

If the surface admits no flat strip, then Ω=ΩNF=T1M,
so that the result follows from theorem 1.3.
It remains to show the following result.

Proposition 3.1

Let M be a compact connected orientable nonpositively curved surface.
If it admits a nonperiodic flat strip,
then it admits also a periodic flat strip.

The proof is inspired by unpublished work of Cao and Xavier.
We proceed by contradiction. Assume that M is not a flat torus,
but admits a nonperiodic flat strip.

There is an isometric embedding from I×R+ to the universal cover
˜M. The interval I is necessarily bounded. Indeed, the manifold
M is compact so it admits a relatively compact connected fundamental domain
for the action of its fundamental group on ˜M. Such a fundamental
domain cannot be completely included in a flat strip with infinite width.

Let Rmax∈R+ be the supremum of the widths r
of nonperiodic flat strips [0,r]×R+
of ˜M. The above argument shows that Rmax is finite.

Consider a vector ~v∈T1˜M generating a trajectory
(gt~v)t≥0 on T1˜M that is tangent to a
nonperiodic flat strip [0,R]×R, for some 910Rmax≤R≤Rmax. Assume that R is maximal among the width of all flat
strips containing ~v, and relocate ~v on the boundary
of the strip. Assume also that the trajectory (gt~v)t≥0
bounds the right side {R}×R+ of the flat strip and denote by v
the image of ~v on T1M.

Since M is compact, we can assume that there is a subsequence gtnv,
with tn→+∞, such that gtnv converges to some vector
v∞. This vector also lies on a flat strip of width at least R.
Indeed, consider a lift ~v∞ of v∞ and isometries
γn of ~M such that γn(gtn~v) converges to
~v∞. Every point on the half-ball of radius R centered on the
base point of ~v∞ is accumulated by points on the euclidean
half-balls centered on γn(gtn~v), so the curvature vanishes
on that half-ball. We can talk about the segment in the half-ball starting
from the base point of ~v∞ and orthogonal to the trajectory of
~v∞. Vectors based on that segment and parallel to
~v∞ are accumulated by vectors generating geodesics in the flat
strips bounding γn(gtn~v). Hence the curvature vanishes
along the geodesics starting from these vectors and we get a flat strip of
width at least R.

If v∞ lies on a periodic flat strip, the proof is finished.
Assume therefore that the flat strip of v∞ is not periodic.

The idea is now to use the nonperiodic flat strip of v to
construct a flat strip of width strictly larger than Rmax.
By definition of Rmax, this new flat strip is necessarily periodic,
and we get the desired result.

The vectors gtnv converges to v∞, so consider t>0
so that the base point of gtv is very close to the base point of
v∞ and the image of gtv by the parallel transport from
T1π(gtv)M to T1π(v)M makes a small angle θ with v.
Observe that this angle θ is nonzero. Indeed, otherwise,
the flat strips bounded by γn(gtn~v) and ~v∞
would be parallel. The flat strip bounded by ~v∞ would
extend the flat strip bounded by γn(gtn~v) by a quantity
roughly equal to the distance between their base points, ensuring that the
flat strip bounded by ~v is actually larger than R and
contradicting the fact that R is the width of this flat strip.

Let us now consider a time t such that gtv is close to v∞,
with the angle between these two vectors denoted by θ>0.
When the flat strip comes back close to v∞ at time t, it cuts
the boundary of the flat strip along a segment whose length is denoted by L.
Without loss of generality, we may assume
that v∞ lies on the right boundary of its flat strip.
Let us consider the highest rectangle of length L/2 that we can
put at the end of this segment, on its right side, and that belongs
to the returning flat strip of gt(v) but not to the flat strip
of v∞. This rectangle is pictured below, its width is denoted by H.

The quantities H and L can be computed
using elementary euclidean trigonometry.

H=R2cosθ≥R2

Misplaced \cr

The same computation works when v∞ is not on the boundary
of its flat strip. The rectangle has a width bounded from below and a length
going to infinity when θ goes to zero.

Every time the trajectory of v comes back near v∞,
we get a new rectangle, and this gives
a sequence of rectangles of increasing length right next to the flat strip.
The next picture shows these rectangles in the universal cover ˜M.

These rectangles are tangent to the flat strip of width R bounded by
v∞, so that we get a sequence of flat rectangles of width
3R/2 and length L going to infinity. By compactness of M, this sequence
accumulates to some infinite flat strip of width 3R/2>Rmax. Therefore
this flat strip is periodic, by definition of Rmax. This ends the proof.

This section is devoted to the proof of
proposition 2.4
and theorem 1.3.

4.1 Closing lemma, local product structure and transitivity

Let X be a metric space, and (ϕt)t∈R be a continuous flow acting
on X. In this section, we recall three fundamental dynamical properties
that we use in the sequel: the closing lemma, the local product structure,
and transitivity.

When these three properties are satisfied on X,
we proved in [CS10] (prop. 3.2 and corollary 2.3) that
the conclusion of proposition 2.4
holds on X: the invariant probability measures
supported by periodic orbits are dense
in the set of all Borel invariant probability measures on X.

In [Pa61], Parthasarathy notes that the density of Dirac measures on
periodic orbits is important to understand the dynamical properties of the
invariant probability measures, and he asks under which assumptions it is
satisfied.
In the next sections, we will prove weakened versions of these three properties
(closing lemma, local product and transitivity),
and deduce proposition 2.4.

Definition 4.1

A flow ϕt on a metric space X
satisfies the closing
lemma if for all points v∈X, and ε>0,
there exist a neighbourhood
V of v , δ>0 and a t0>0
such that for all w∈V and all t>t0 with d(w,ϕtw)<δ and
ϕtw∈V,
there exists p0 and l>0, with |l−t|<ε,
ϕlp0=p0, and d(ϕsp0,ϕsw)<ε
for 0<s<min(t,l).

Definition 4.2

The flow ϕt is said to admit a
local product structure if all points u∈X have
a neighbourhood V which satisfies : for all
ε>0, there exists a positive constant δ,
such that for all v,w∈V with d(v,w)≤δ,
there is a point <v,w>∈X,
a real number t with |t|≤ε, so that:

<v,w>∈Wsuε(ϕt(v))∩Wssε(w).

Definition 4.3

The flow (ϕt)t∈R is transitive if for all non-empty
open sets U and V of X, and T>0, there is t≥T such that
ϕt(U)∩V≠∅.

Recall that if X is a Gδ subset of a
complete separable metric space, then it is a Polish space, and the set
M1(X) of invariant probability measures on X is also a Polish
space. As a result, the Baire theorem holds on this space [Bi99]
th 6.8. In particular, this will be the case for the set X=ΩNF
when it is open in Ω, since Ω is a closed subset of T1M.

If M is negatively curved, we saw in [CS10] that the restriction
of (gt)t∈R to Ω
satisfies the closing lemma, the local product structure, and is transitive.
Note that we do not need any (lower or upper) bound on the curvature, i.e.
we allow the curvature to go to 0 or to −∞
in some noncompact parts of M. In particular, the conclusions of all
theorems of this article apply to the geodesic flow on
the nonwandering set of any nonelementary negatively curved manifold.

4.2 Closing lemma and transitivity on ΩNF

We start by a proposition essentially due to G. Knieper
([K98] prop 4.1).

Proposition 4.4

Let v∈ΩNF be a recurrent
vector which does not bound a flat strip. Then v∈Ωhyp, i.e.
its strong stable (resp. unstable) manifold coincides
with its stable (resp. unstable) horosphere.

Proof :

Let ˜M the universal cover of M and
~v∈T1˜M be a lift of v.
Assume that there exists w∈T1M which belongs to the stable horosphere,
but not to the strong stable manifold of v.
We can therefore find c>0,
such that 0<c≤d(gt~v,gt~w)≤d(v,w), for all t≥0.
Let us denote by Γ the deck transformation group of the
covering ~M→M.
This group acts by isometries on T1~M.
The vector v is recurrent, so there exists γn∈Γ,
tn→∞, with γn(gtn~v)→~v.
Therefore, for all s≥−tn,
we have c≤d(gtn+s~v,gtn+s~w)=d(gsγngtnv,gsγngtnw)≤d(v,w).
Up to a subsequence, we can assume that γngtnw
converges to a vector z.
Then we have for all s∈R, 0<c≤d(gs~v,gsz)≤d(v,w).
The flat strip theorem shows that ~v bounds a flat strip
(see e.g. [Ba95] cor 5.8).
This concludes the proof.
□

In order to state the next result, we recall a definition.
The ideal boundary of the universal cover, denoted by ∂~M,
is the set of equivalent classes of half geodesics that stay at a bounded
distance of each other, for all positive t. We note u+ the class
associated to the geodesic t↦u(t), and u− the class
associated to the geodesic t↦u(−t).

Lemma 4.5 (Weak local product structure)

Let M be a complete, connected, nonpositively curved manifold,
and v0 be a vector that does not bound a flat strip.

For all ε>0, there exists
δ>0, such that if v,w∈T1M satisfy d(v,v0)≤δ,
d(w,v0)≤δ, there exists a vector u=<v,w>
satisfying u−=v−, u+=v+, and d(u,v0)≤ε.

Moreover, if v,w∈Thyp,
then u=<v,w>∈Thyp.

This lemma will be applied later
to recurrent vectors that do not bound a flat strip;
these are all in Ωhyp.

Proof :

The first item of this lemma is an immediate reformulation
of [Ba95] lemma 3.1 page 50.
The second item comes from the definition of the set Thyp
of vectors whose stable (resp. unstable)
manifold coincide with the stable (resp. unstable) horosphere.
□

Note that a priori, the local product structure as stated in definition
4.2 and in [CS10]
is not satisfied on ΩNF: if v,w are in ΩNF,
the local product <v,w>
does not necessarily belong to ΩNF.

Lemma 4.6

Let M be a nonpositively curved manifold such that
ΩNF is open in Ω.
Then the closing lemma (see definition 4.1)
is satisfied in restriction to ΩNF.

Proof :

We adapt the argument of Eberlein [E96]
(see also the proof of theorem 7.1 in [CS10]).
Let u∈ΩNF, ε>0 and U be a neighborhood
of u in Ω. We can assume that
U⊂ΩNF⊂Ω
since ΩNF is open in Ω.
Given v∈U∩ΩNF, with d(gtv,v) very small
for some large t, it is enough to find a periodic orbit p0∈U
shadowing the orbit of v during a time t±ε.
Since the sets Ωhyp and ΩNF
have the same periodic orbits, we will deduce that
p0∈Ωhyp⊂ΩNF.

Choose ε>0, and assume by contradiction
that there exists a sequence (vn) in ΩNF, vn→u,
and tn→+∞, such that
d(vn,gtnvn)→0, with no periodic orbit of length
approximatively tn shadowing the orbit of vn.

Lift everything to T1~M. There exists ε>0,
~vn→~u, tn→+∞,
and a sequence of isometries φn of ˜M s.t.
d(~vn,dφn∘gtn~vn)→0.
Now, we will show that for n large enough, φn is an
axial isometry, and find on its axis a
vector ~pn which is the lift of a periodic orbit of
length ωn=tn±ε shadowing the orbit of vn.
This will conclude the proof by contradiction.

Let γ~u be the geodesic determined by
~u, and u± its endpoints at infinity,
x∈~M (resp. xn, yn) the
basepoint of ~u (resp. ~vn, gtn~vn).
As ~vn→~u, tn→+∞,
xn→x, and d(φ−1n(xn),yn)→0,
we see easily that φ−1n(x)→u+. Similary, φn(x)→u−.

Since ~u does not bound a flat strip, Lemma 3.1
of [Ba95] implies that for all α>0,
there exist neighbourhoods Vα(u−) and Vα(u+)
of u− and u+ respectively, in the boundary at
infinity of ~M, such that for all ξ−∈Vα(u−)
and ξ+∈Vα(u+),
there exists a geodesic joining ξ− and ξ+ and
at distance less than α from x=γ~u(0).

Choose α=ε/2.
We have φn(x)→u− and φ−1n(x)→u+, so for
n large enough,
φn(Vε/2(u−))⊂Vε/2(u−)
and φ−1n(Vε/2(u+))⊂Vε/2(u+).
By a fixed point argument, we find two fixed points
ξ±n∈Vε/2(u±) of φn,
so that φn is an axial isometry.

Consider the geodesic joining ξ−n to ξ+n given
by W. Ballmann’s lemma.
It is invariant by φn, which acts by translation on it,
so that it induces on M a periodic geodesic, and on
T1M a periodic orbit of the geodesic flow.
Let pn be the vector of this orbit minimizing the distance
to u, and ωn its period.
The vector pn is therefore close to vn, and its period close to tn,
because dφ−1n(~pn)=gωn~pn
projects on T1M to pn,
dφ−1n(~vn)=gtn~vn projects to
gtnvn, d(gtnvn,vn) is small,
and φn is an isometry.
Thus, we get the desired contradiction.
□

Lemma 4.7 (Transitivity)

Let M be a connected, complete, nonpositively curved manifold
which contains at least three distinct periodic orbits
that do not bound a flat strip.
If ΩNF is open in Ω,
then the restriction of the geodesic flow to any of the two sets
ΩNF or Ωhyp is transitive.

Transitivity of the geodesic flow on Ω
was already known under the so-called duality condition,
which is equivalent to the equality Ω=T1M
(see [Ba95] for details and references).
In that case, Ωhyp is dense in T1M.

Proof :

Let U1 and U2 be two open sets in ΩNF.
Let us show that there is a trajectory in ΩNF that starts from
U1 and ends in U2. This will prove transitivity on ΩNF.

The closing lemma implies that periodic orbits
in Ωhyp are dense in ΩNF and Ωhyp.
So we can find two periodic vectors
v1 in Ωhyp∩U1, and v2 in Ωhyp∩U2.
Let us assume that v2 is not
opposite to v1 or an iterate of v1:
−v2∉∪t∈Rgt({v1}).
Then there is a vector v3∈T1M
whose trajectory is negatively asymptotic to the trajectory of
v1 and positively asymptotic to
the trajectory of v2, cf [Ba95] lemma 3.3.
Since v1 and v2 are in Ωhyp,
the vector v3 also belongs to Thyp,
and therefore does not bound a periodic flat strip.

Let us show that v3 is nonwandering.
First note that there is also a trajectory negatively asymptotic to
the negative trajectory of v2 and positively asymptotic to
the trajectory of v1. That is, the two periodic orbits v1, v2
are connected as pictured below.

This implies that the two connecting orbits are nonwandering:
indeed, using the local product structure, we can glue the two
connecting orbits to obtain a trajectory that starts close to v3,
follows the second connecting orbit, and then follows the orbit of
v3, coming back to the vector v3 itself. Hence v3 is in
Ω. Since it is in Thyp
it belongs to Ωhyp⊂ΩNF
and we are done.

If v1 and v2 generate opposite trajectories, then we take a third
periodic vector w that does not bound a flat strip,
and connect first v1 to w then w to v2.
Using again the product structure, we can glue the connecting
orbits to create a nonwandering trajectory from U1 to U2.
□

Remark 4.8

We note
that without any topological assumption on ΩNF,
the same argument gives transitivity of the geodesic flow
on the closure of the set of periodic hyperbolic vectors.

4.3 Density of Dirac measures on periodic orbits

Let M be a connected, complete, nonpositively curved manifold,
which admits at least three different periodic orbits that do not
bound a flat strip. Assume
that ΩNF is open in Ω.
Then the Dirac measures supported by the periodic orbits
of the geodesic flow (gt)t∈R that are in ΩNF,
are dense in the set of all invariant probability measures defined
on ΩNF.

Proof :

We first show that Dirac measures on periodic orbits not bounding
a flat strip are dense in the set of ergodic invariant probability measures
on ΩNF.

Let μ be an ergodic invariant probability measure supported
by ΩNF. By Poincaré and Birkhoff theorems,
μ-almost all vectors are recurrent and generic w.r.t. μ.
Let v∈ΩNF be such a recurrent generic vector w.r.t. μ
that belongs to ΩNF. The closing lemma 4.6
gives a periodic orbit close to v.
Since ΩNF is open in Ω,
that periodic orbit is in fact in ΩNF.
The Dirac measure on that orbit is close to μ and the claim is proven.

The set M1(Ω) is the convex hull of the
set of invariant ergodic probability measures,
so the set of convex combinations of periodic
measures not bounding a flat strip is dense in the set of
all invariant probability measures on ΩNF.
It is therefore enough to prove that periodic
measures not bounding a flat strip are dense in the set of convex
combinations of such measures.
The argument follows [CS10], with some subtle differences.

Let x1, x3, …, x2n−1 be periodic vectors of ΩNF
with periods l1, l3,…, l2n−1, and c1, c3,…, c2n−1
positive real numbers with Σc2i+1=1. Let us denote the
Dirac measure on the orbit of a periodic vector p by
δp. We want to find a periodic vector p such that
δp is close to the sum Σc2i+1δx2i+1.
The numbers c2i+1 may be assumed to be rational numbers of the
form p2i+1/q. Recall that the xi are in fact in Ωhyp.

The flow is transitive on ΩNF (lemma 4.7),
hence for all i, there is a vector x2i∈ΩNF
close to x2i−1 whose trajectory becomes close to x2i+1,
say, after time t2i. We can also find a point x2n close to
x2n−1 whose trajectory becomes close to x1 after some time.
The proof of lemma 4.7 actually tells us that the
x2i can be chosen in Ωhyp.

Now these trajectories can be glued together,
using the local product on Ωhyp
(lemma 4.5) in the neighbourhood
of each x2i+1∈Ωhyp, as follows:
we fix an integer N, large enough.
First glue the piece of periodic orbit starting
from x1, of length Nl1p1, together with the orbit of x2, of
length t2. The resulting orbit ends in a neighbourhood of x3, and
that neighbourhood does not depend on the value of N. This orbit is
glued with the trajectory starting from x3, of length Nl2p2,
and so on (See [C04] for details).

We end up with a vector close to x1, whose trajectory is
negatively asymptotic to the trajectory of x1, then turns Np1 times
around the first periodic orbit, follows the trajectory of
x2 until it reaches x3; then it turns Np3 times around the
second periodic orbit, and so on, until it reaches x2n and goes
back to x1, winding up on the trajectory of x1.
The resulting trajectory is in Thyp and,
repeating the argument from Lemma 4.7,
we see that it is nonwandering.

Finally, we use the closing lemma on ΩNF to
obtain a periodic orbit in ΩNF.
When N is large, the time spent going from one periodic orbit to another
is small with respect to the time winding up around the periodic orbits,
so the Dirac measure on the resulting periodic orbit is close to the sum
∑ic2i+1δx2i+1 and the theorem is proven.

□

The proof of theorem 1.3 is then straightforward
and follows verbatim from the arguments given in [CS10].
We sketch the proof for the comfort of the reader.

Proof :

Proposition 2.4 ensures that
ergodic measures are dense in the set of probability
measures on ΩNF.
The fact that they form a Gδ-set is well known.

The fact that invariant measures of full support are a dense Gδ-subset
of the set of invariant probability measures on ΩNF is
a simple corollary of the density of periodic orbits
in ΩNF, which itself follows from the closing lemma.

Finally, the intersection of two dense Gδ-subsets
of M1(ΩNF) is
still a dense Gδ-subset of M1(ΩNF),
because this set has the Baire property.
This concludes the proof.
□

4.4 Examples

We now build examples for which the hypotheses or results
presented in that article do not hold.

We start by an example of a surface for which ΩNF is not open in
Ω. First we consider a surface made up of
an euclidean cylinder put on an euclidean plane. Such surface is built
by considering an horizontal line and a vertical line
in the plane, and connecting them with a convex arc that is
infinitesimally flat at its ends.
The profile thus obtained is then rotated along the vertical axis.
The negatively curved part is greyed in the figure below.

We can repeat that construction so as to line up cylinders
on a plane. Let us use cylinders of the same size and shape,
and take them equally spaced.
The quotient of that surface by the natural Z-action
is a pair of pants, its three ends being euclidean flat cylinders.

These cylinders are bounded by three closed geodesics that are
accumulated by points of negative curvature.
The nonwandering set of the Z-cover is the inverse image
of the nonwandering set of the pair of pants.
As a result, the lift of the three closed geodesics to the Z-cover
are nonwandering geodesics. They are in fact accumulated by
periodic geodesics turning around the cylinders a few times
in the negatively curved part, cf [CS11], th. 4.2 ff.
We end up with a row of cylinders on a strip bounded by two
nonwandering geodesics. These are the building blocks for our
example.

We start from an euclidean half-plane and pile up alternatively
rows of cylinders with bounding geodesics γi and γ′i,
and euclidean flat strips. We choose the width so that
the total sum of the widths of all strips is converging.
We also increase the spacing between the cylinders from
one strip to another so as to insure that they do not accumulate
on the surface.
The next picture is a top view of our surface, cylinders appear
as circles.

All the strips accumulate on a geodesic γ∞
that is nonwandering because it is in the closure of the periodic geodesics.
We can insure that it does not bound a flat strip by
mirroring the construction on the other side of γ∞.
So γ∞ is in ΩNF, and is approximated by
geodesics γi that belong to Ω and bound a flat strip.
Thus, ΩNF is not open in Ω.
We conjecture that ergodicity is a generic property in the set
of all probability measures invariant by the geodesic flow on that surface.
The flat strips should not matter here
since they do not contain recurrent trajectories, but our method does not apply
to that example.

The next example, due to Gromov [Gr78], is detailed in
[Eb80] or [K98]. Let T1 be a torus with one
hole, whose boundary is homeomorphic to S1, endowed with a nonpositively
curved metric, negative far from the boundary, and zero on a flat cylinder
homotopic to the boundary. Let M1=T1×S1. Similarly, let T2 be the
image of T1 under the symmetry with respect to a plane containing ∂T1, and M2=S1×T2. The manifolds M1 and M2 are
3-dimensional manifolds whose boundary is a euclidean torus. We glue them
along this boundary to get a closed manifold M which contains around the
place of gluing a thickened flat torus, isometric to [−r,r]×T2, for some r>0.

Consider the flat 2-dimensional torus {0}×T2 embedded in
M. Choose an irrational direction {θ} on its unit tangent bundle
and lift the normalized Lebesgue measure of the flat torus to the invariant
set of unit tangent vectors pointing in this irrational direction θ.
This measure is an ergodic invariant probability measure on T1M, and the
argument given in [CS11] shows that it is not in the closure of the set of
invariant ergodic probability measures of full support. In particular, ergodic
measures are not dense, and therefore not generic. Note also that this
measure is in the closure of the Dirac orbits supported by periodic orbits
bounding flat strips (we just approximate θ by a rational number),
but cannot be approximated by Dirac orbits
on periodic trajectories that do not bound flat strips.

This does not contradict our results though,
because this measure is supported in Ω∖ΩNF
(which is closed).

5.1 Measure-theoretic entropy

Let X be a Polish space, (ϕt)t∈R a continuous flow on X,
and μ a Borel invariant probability measure on X.
As the measure theoretic entropy satisfies the relation
hμ(ϕt)=|t|hμ(ϕ1),
we define here the entropy of the application T:=ϕ1.

Definition 5.1

Let P={P1,…,PK}
be a finite partition of X into Borel sets.
The entropy of the partition P is the quantity

Hμ(P)=−∑P∈Pμ(P)logμ(P).

Denote by ∨n−1i=0T−iP the finite
partition into sets of the form
Pi1∩T−1Pi2∩⋯∩T−n+1Pin.
The measure theoretic entropy of T=ϕ1 w.r.t.
the partition P is defined by the limit

Proposition 5.2

Let (Pk)k∈N
be a increasing sequence of finite partitions of X
into Borel sets such that
∨∞k=0Pk generates the Borel σ-algebra of X.
Then the measure theoretic entropy of ϕ1 satisfies

hμ(ϕ1)=supk∈Nhμ(ϕ1,Pk).

5.2 Generic measures have zero entropy

Theorem 5.3

Let M be a
connected, complete, nonpositively curved manifold,
whose geodesic flow admits at
least three different periodic orbits, that do not bound a flat strip.
Assume that ΩNF is open in Ω.
The set of invariant probability measures on ΩNF with zero
entropy is a dense Gδ subset of the
set M1(ΩNF) of invariant probability
measures supported in ΩNF.

Recall here that on a nonelementary negatively curved manifold,
Ω=ΩNF so that the above theorem applies on the full
nonwandering set Ω.

The proof below is inspired from the proof of
Sigmund [Si70], who treated
the case of Axiom A flows on compact manifolds, and from results of
Abdenur, Bonatti, Crovisier [ABC10]
who considered nonuniformly hyperbolic diffeomorphisms on compact manifolds.
But no compactness assumption is needed in our statement.

Proof :

Remark first that on any Riemannian manifold M,
if B=B(x,r) is a small ball, r>0 being strictly less
than the injectivity radius of M at the point x, any geodesic
(and in particular any periodic geodesic) intersects the boundary of B
in at most two points.
Lift now the ball B to the set T1B
of unit tangent vectors of T1M with base points in B.
Then the Dirac measure supported on
any periodic geodesic intersecting B
gives zero measure to the boundary of T1B.

Choose a countable family of balls Bi=B(xi,ri), with centers dense in
M. Subdivide each lift T1Bi on the unit tangent bundle T1M into
finitely many balls, and denote by (Bj) the countable family of
subsets of T1M that we obtain. Any finite family of such sets
Bj induces a finite partition of ΩNFP into Borel sets
(finite intersections of the Bj’s, or their complements ). Denote
by Pk the finite partition induced by the finite family of sets
(Bj)0≤j≤k. If the family Bj is well chosen,
the increasing sequence (Pk)k∈N is such that
∨∞k=0Pk generates
the Borel σ-algebra.

Set X=ΩNF. According to proposition
2.4, the family D of Dirac measures
supported on periodic orbits of X is dense in M1(X).
Denote by M1Z(X) the subset of probability measures
with entropy zero in M1(X).
The family D of Dirac measures
supported on periodic orbits of X is included
in M1Z(X), is dense in M1(X),
satisfies μ(∂Pk)=0 and
hμ(Pk)=0 for all k∈N and μ∈D.

Fix any μ0∈D.
Note that the limit in (1) always exists, so
that it can be replaced by a liminf.
As μ0 satisfies μ0(∂Pk)=0,
if a sequence
μi∈M1(X) converges in the
weak topology to μ0,
it satisfies for all n∈N,
Hμi(∨nj=0g−jPk)→Hμ0(∨nj=0g−jPk)
when i→∞.
In particular, the set

{μ∈M1(X),Hμ(∨nj=0g−jPk)<Hμ0(∨nj=−ngjPk)+1r},

for r∈N∗, is an open set.
We deduce that MZ(X) is
a Gδ-subset of M(X).
Indeed,

M1Z(X)

=

{μ∈M1(X),hμ(g1)=0=hμ0(g1)}

=

∩k∈N{μ∈M1(X),hμ(g1,Pk)=0=hμ0(g1,Pk)}

=

∩k∈N∩∞r=1{μ∈M1(X),0≤hμ(g1,Pk)<1r=hμ0(g1,Pk)+1r}

=

∩k∈N∩∞r=1∩∞m=1∪∞n=m

{μ∈M1(X),1n+1Hμ(∨nj=0g−jPk)<1n+1Hμ0(∨nj=0g−jPk)+1r}.

The fact that M1Z(X) is dense is obvious
because it contains the family D of periodic orbits of X.
□

6.1 Topological mixing

Let (ϕt)t∈R be a continuous flow on a Polish space X.
The flow is said topologically mixing
if for all open subsets U,V of X,
there exists T>0, such that for all t≥T,
ϕtU∩V≠∅.
This property is of course stronger than transitivity:
the flow is transitive if
for all open subsets U,V of X,
and all T>0, there exists t≥T, ϕtU∩V≠∅.
An invariant measure μ under the flow
is strongly mixing if for all Borel sets A and B we have
μ(A∩ϕtB)→μ(A)μ(B) when t→+∞.

An invariant measure cannot be strongly mixing if the flow itself
is not topologically mixing on its support (see e.g. [W82]).
We recall therefore some results about topological mixing,
which are classical on negatively curved manifolds, and still true here.

Let M be a connected rank one manifold,
such that all tangent vectors are nonwandering (Ω=T1M).
Then the geodesic flow is topologically mixing.

Also related is the work of M. Babillot [Ba01]
who obtained the mixing of the measure of maximal entropy under
suitable assumptions, with the help of a geometric cross ratio.

Proposition 6.2

Let M be a connected, complete, nonpositively curved manifold,
whose geodesic flow admits at least three distinct periodic orbits,
that do not bound a flat strip.
If ΩNF
is open in Ω, then
the restriction of the geodesic flow to ΩNF
is topologically mixing iff the length spectrum
of the geodesic flow restricted to ΩNF
is non arithmetic.

Proof :

Assume first that the geodesic flow restricted to ΩNF
is topologically mixing. The argument is classical.
Let u∈ΩNF be a vector, and ε>0.
Let δ>0 and U⊂ΩNF
be a neighbourhood of u of the form U=B(u,δ)∩ΩNF
where the closing lemma is satisfied (see lemma 4.6).

Topological mixing on ΩNF
implies that there exists T>0,
s.t. for all t≥T, gtU∩U≠∅.
Thus, for all t≥T there exists v∈U∩gtU,
so that d(gtv,v)≤δ.

We can apply the closing lemma to v,
and obtain a periodic orbit of ΩNF
of length t±ε shadowing the orbit of v during the time t.
As it is true for all ε>0 and large t>0,
it implies the non arithmeticity of the length spectrum of the geodesic flow
in restriction to ΩNF.

We assume now that the length spectrum of the geodesic
flow restricted to ΩNF is non arithmetic
and we show that the geodesic flow is topologically mixing.
In [D00], she proves this implication on negatively
curved manifolds, by using intermediate properties
of the strong foliation.
We give here a direct argument.

∙ First, observe that it is enough to prove that for any
open set U∈ΩNF, there exists T>0,
such that for all t≥T, gtU∩U≠∅.
Indeed, if U, V are two open sets of ΩNF,
by transitivity of the flow, there exists u∈U
and T0>0 s.t. gT0u∈V. Now, by continuity of
the geodesic flow, we can find a neighbourhood
U′ of u in U, such that gT0(U′)⊂V.
If we can prove that for all large t>0,
gt(U′)∩U′≠∅, we obtain that for
all large t>0, gtU∩V≠∅.

∙ Fix an open set U⊂ΩNF.
Periodic orbits of Ωhyp are dense in ΩNF.
Choose a periodic orbit
p∈U∩Ωhyp. As U is open, there exists ε>0,
such that gtp∈U, for all t∈[−3ε,3ε].
By non arithmeticity of the length spectrum,
there exists another periodic vector p0∈Ωhyp, and positive integers
n,m∈Z, |nl(p)−ml(p0)|<ε.
Assume that 0<nl(p)−ml(p0)<ε.

∙ By transitivity of the geodesic flow on ΩNF,
and local product
choose a vector v negatively asymptotic to the negative geodesic
orbit of p and positively asymptotic to the geodesic orbit of p0,
and a vector w negatively asymptotic to the orbit of p0
and positively asymptotic
to the orbit of p. By lemma 4.5 (2),
v and w are in Thyp. Moreover,
they are nonwandering by the same argument as in the proof
of lemma 4.7.
Using the local product structure and the closing lemma, we can construct
for all
positive integers k1,k2∈N∗ a periodic vector pk1,k2 at distance
less than ε of p, whose orbit turns k1
times around the orbit of p, going from an ε-neighbourhood
of p to an ε-neigbourhood of p0,
with a “travel time” τ1>0, turning around the orbit of p0k2 times, and coming back to the ε-neighbourhood of p,
with a travel time τ2.
Moreover, τ1 and τ2 are independent of k1,k2
and depend only on ε, and on the initial choice of v and w.
The period of pk1,k2 is
k1l(p)+k2l(p0)+C(τ1,τ2,ε),
where C is a constant, and gτpk1,k2 belongs to U
for all τ∈]−ε,ε[.

∙ Now, by non arithmeticity, there exists T>0 large enough, s.t.
the set
Extra open brace or missing close brace
is ε